### Learning Objectives

- Define and use the zero exponent rule
- Define and use the negative exponent rule

## Define and use the zero exponent rule

When we defined the quotient rule, we only worked with expressions like the following: [latex]\frac{{{4}^{9}}}{{{4}^{4}}}[/latex], where the exponent in the numerator (up) was greater than the one in the denominator (down), so the final exponent after simplifying was always a positive number, and greater than zero. In this section, we will explore what happens when we apply the quotient rule for exponents and get a negative or zero exponent.

## What if the exponent is zero?

To see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of 1.

[latex]\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex]

If we were to simplify the original expression using the quotient rule, we would have

[latex]\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex]

If we equate the two answers, the result is [latex]{t}^{0}=1[/latex]. This is true for any nonzero real number, or any variable representing a real number.

The sole exception is the expression [latex]{0}^{0}[/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or DNE (Does Not Exist).

### Exponents of 0 or 1

Any number or variable raised to a power of 1 is the number itself.

[latex]n^{1}=n[/latex]

Any non-zero number or variable raised to a power of 0 is equal to 1

[latex]n^{0}=1[/latex]

The quantity [latex]0^{0}[/latex] is undefined.

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.

### Example

Evaluate [latex]2x^{0}[/latex] if [latex]x=9[/latex]

### Example

Simplify [latex]\frac{{c}^{3}}{{c}^{3}}[/latex].

In the following video there is an example of evaluating an expression with an exponent of zero, as well as simplifying when you get a result of a zero exponent.

## Define and use the negative exponent rule

We proposed another question at the beginning of this section. Given a quotient like [latex] \displaystyle \frac{{{2}^{m}}}{{{2}^{n}}}[/latex] what happens when *n* is larger than *m*? We will need to use the *negative rule of exponents* to simplify the expression so that it is easier to understand.

Let’s look at an example to clarify this idea. Given the expression:

[latex]\frac{{h}^{3}}{{h}^{5}}[/latex]

Expand the numerator and denominator, all the terms in the numerator will cancel to 1, leaving two *h*s multiplied in the denominator, and a numerator of 1.

We could have also applied the quotient rule from the last section, to obtain the following result:

[latex]\begin{array}{r}\frac{h^{3}}{h^{5}}\,\,\,=\,\,\,h^{3-5}\\\\=\,\,\,h^{-2}\,\,\end{array}[/latex]

Putting the answers together, we have [latex]{h}^{-2}=\frac{1}{{h}^{2}}[/latex]. This is true when *h*, or any variable, is a real number and is not zero.

### The Negative Rule of Exponents

For any nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states that

Let’s looks at some examples of how this rule applies under different circumstances.

### Example

Evaluate the expression [latex]{4}^{-3}[/latex].

### Example

Write [latex]\frac{{\left({t}^{3}\right)}}{{\left({t}^{8}\right)}}[/latex] with positive exponents.

### Example

Simplify [latex]{\left(\frac{1}{3}\right)}^{-2}[/latex].

### Example

Simplify.[latex]\frac{1}{4^{-2}}[/latex] Write your answer using positive exponents.

In the follwoing video you will see examples of simplifying expressions with negative exponents.